| L(s) = 1 | − 9-s + 6·19-s + 8·29-s − 14·31-s + 12·41-s + 13·49-s + 20·59-s + 2·61-s − 28·71-s − 16·79-s + 81-s + 12·101-s − 30·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s − 6·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.37·19-s + 1.48·29-s − 2.51·31-s + 1.87·41-s + 13/7·49-s + 2.60·59-s + 0.256·61-s − 3.32·71-s − 1.80·79-s + 1/9·81-s + 1.19·101-s − 2.87·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s − 0.458·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.285793587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285793587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.619447454021176535240103836741, −7.88327540274462329163720754482, −7.87083354399563097921701828962, −7.29141625906304372419516102056, −7.07675014168525748225024065445, −6.83853335680882736108120863477, −6.23717585027297662071734766051, −5.69203359317660362302305299846, −5.58133535647118716731890918066, −5.40479078933068485901212790909, −4.76075589202371209914998399297, −4.19235275388135444557192342561, −4.12669772283713863114032247533, −3.47820096279686066187157957895, −3.13380876250333608641315971359, −2.50951536042092462000759569438, −2.42358184763633077055087820008, −1.52738298164492380055805293306, −1.13227821869206985691070567132, −0.45114119390290028447124511973,
0.45114119390290028447124511973, 1.13227821869206985691070567132, 1.52738298164492380055805293306, 2.42358184763633077055087820008, 2.50951536042092462000759569438, 3.13380876250333608641315971359, 3.47820096279686066187157957895, 4.12669772283713863114032247533, 4.19235275388135444557192342561, 4.76075589202371209914998399297, 5.40479078933068485901212790909, 5.58133535647118716731890918066, 5.69203359317660362302305299846, 6.23717585027297662071734766051, 6.83853335680882736108120863477, 7.07675014168525748225024065445, 7.29141625906304372419516102056, 7.87083354399563097921701828962, 7.88327540274462329163720754482, 8.619447454021176535240103836741
